Let $D$ be an open set in $\mathbb{R} \times \mathbb{R}^{n}$. Let $f_{m}$ be a sequence in $C\left(D ; \mathbb{R}^{n}\right)$ that converges uniformly on compact subset of $D$ to $f_{0} \in C\left(D ; R^{n}\right)$. Let $\left(t_{m}, a_{m}\right)$ be a sequence in $D$ such that $$ \lim _{m \rightarrow \infty}\left(t_{m}, a_{m}\right)=\left(t_{0}, a_{0}\right) $$ for some $\left(t_{0}, a_{0}\right) \in D .$ For each $m \in \mathbb{N} \cup\{0\}$ let $x_{m}$ be a solution of $$ \left\{\begin{array}{l} \frac{d z}{d t}=f_{m}(t, x) \\ x\left(t_{m}\right)=a_{m} \end{array}\right. $$ I have to show that there exists a $\delta>0$ and $N_{1} \in \mathbb{N}$ such that $$ \left[ t_{0}-\delta, t_{0}+\delta\right] \subset I\left(x_{m}\right) $$ for all $m \geq N_{1}$, where $I\left(x_{m}\right)$ denotes the maximal interval of existence for $x_{m}$.
Can someone help me to solve this problem. I don't know how to approach or what theorem to use for this problem.